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<div class="titlepage"><div><div><h4 class="title">
<a name="math_toolkit.dist_ref.dists.rayleigh"></a><a class="link" href="rayleigh.html" title="Rayleigh Distribution">Rayleigh Distribution</a>
</h4></div></div></div>
<pre class="programlisting"><span class="preprocessor">#include</span> <span class="special">&lt;</span><span class="identifier">boost</span><span class="special">/</span><span class="identifier">math</span><span class="special">/</span><span class="identifier">distributions</span><span class="special">/</span><span class="identifier">rayleigh</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">&gt;</span></pre>
<pre class="programlisting"><span class="keyword">namespace</span> <span class="identifier">boost</span><span class="special">{</span> <span class="keyword">namespace</span> <span class="identifier">math</span><span class="special">{</span>

<span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">RealType</span> <span class="special">=</span> <span class="keyword">double</span><span class="special">,</span>
          <span class="keyword">class</span> <a class="link" href="../../../policy.html" title="Chapter 22. Policies: Controlling Precision, Error Handling etc">Policy</a>   <span class="special">=</span> <a class="link" href="../../pol_ref/pol_ref_ref.html" title="Policy Class Reference">policies::policy&lt;&gt;</a> <span class="special">&gt;</span>
<span class="keyword">class</span> <span class="identifier">rayleigh_distribution</span><span class="special">;</span>

<span class="keyword">typedef</span> <span class="identifier">rayleigh_distribution</span><span class="special">&lt;&gt;</span> <span class="identifier">rayleigh</span><span class="special">;</span>

<span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">RealType</span><span class="special">,</span> <span class="keyword">class</span> <a class="link" href="../../../policy.html" title="Chapter 22. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;</span>
<span class="keyword">class</span> <span class="identifier">rayleigh_distribution</span>
<span class="special">{</span>
<span class="keyword">public</span><span class="special">:</span>
   <span class="keyword">typedef</span> <span class="identifier">RealType</span> <span class="identifier">value_type</span><span class="special">;</span>
   <span class="keyword">typedef</span> <span class="identifier">Policy</span>   <span class="identifier">policy_type</span><span class="special">;</span>
   <span class="comment">// Construct:</span>
   <span class="identifier">rayleigh_distribution</span><span class="special">(</span><span class="identifier">RealType</span> <span class="identifier">sigma</span> <span class="special">=</span> <span class="number">1</span><span class="special">)</span>
   <span class="comment">// Accessors:</span>
   <span class="identifier">RealType</span> <span class="identifier">sigma</span><span class="special">()</span><span class="keyword">const</span><span class="special">;</span>
<span class="special">};</span>

<span class="special">}}</span> <span class="comment">// namespaces</span>
</pre>
<p>
          The <a href="http://en.wikipedia.org/wiki/Rayleigh_distribution" target="_top">Rayleigh
          distribution</a> is a continuous distribution with the <a href="http://en.wikipedia.org/wiki/Probability_density_function" target="_top">probability
          density function</a>:
        </p>
<div class="blockquote"><blockquote class="blockquote"><p>
            <span class="serif_italic">f(x; sigma) = x * exp(-x<sup>2</sup>/2 σ<sup>2</sup>) / σ<sup>2</sup></span>
          </p></blockquote></div>
<p>
          For sigma parameter <span class="emphasis"><em>σ</em></span> &gt; 0, and <span class="emphasis"><em>x</em></span>
          &gt; 0.
        </p>
<p>
          The Rayleigh distribution is often used where two orthogonal components
          have an absolute value, for example, wind velocity and direction may be
          combined to yield a wind speed, or real and imaginary components may have
          absolute values that are Rayleigh distributed.
        </p>
<p>
          The following graph illustrates how the Probability density Function(pdf)
          varies with the shape parameter σ:
        </p>
<div class="blockquote"><blockquote class="blockquote"><p>
            <span class="inlinemediaobject"><img src="../../../../graphs/rayleigh_pdf.svg" align="middle"></span>

          </p></blockquote></div>
<p>
          and the Cumulative Distribution Function (cdf)
        </p>
<div class="blockquote"><blockquote class="blockquote"><p>
            <span class="inlinemediaobject"><img src="../../../../graphs/rayleigh_cdf.svg" align="middle"></span>

          </p></blockquote></div>
<h5>
<a name="math_toolkit.dist_ref.dists.rayleigh.h0"></a>
          <span class="phrase"><a name="math_toolkit.dist_ref.dists.rayleigh.related_distributions"></a></span><a class="link" href="rayleigh.html#math_toolkit.dist_ref.dists.rayleigh.related_distributions">Related
          distributions</a>
        </h5>
<p>
          The absolute value of two independent normal distributions X and Y, √ (X<sup>2</sup> +
          Y<sup>2</sup>) is a Rayleigh distribution.
        </p>
<p>
          The <a href="http://en.wikipedia.org/wiki/Chi_distribution" target="_top">Chi</a>,
          <a href="http://en.wikipedia.org/wiki/Rice_distribution" target="_top">Rice</a>
          and <a href="http://en.wikipedia.org/wiki/Weibull_distribution" target="_top">Weibull</a>
          distributions are generalizations of the <a href="http://en.wikipedia.org/wiki/Rayleigh_distribution" target="_top">Rayleigh
          distribution</a>.
        </p>
<h5>
<a name="math_toolkit.dist_ref.dists.rayleigh.h1"></a>
          <span class="phrase"><a name="math_toolkit.dist_ref.dists.rayleigh.member_functions"></a></span><a class="link" href="rayleigh.html#math_toolkit.dist_ref.dists.rayleigh.member_functions">Member
          Functions</a>
        </h5>
<pre class="programlisting"><span class="identifier">rayleigh_distribution</span><span class="special">(</span><span class="identifier">RealType</span> <span class="identifier">sigma</span> <span class="special">=</span> <span class="number">1</span><span class="special">);</span>
</pre>
<p>
          Constructs a <a href="http://en.wikipedia.org/wiki/Rayleigh_distribution" target="_top">Rayleigh
          distribution</a> with σ <span class="emphasis"><em>sigma</em></span>.
        </p>
<p>
          Requires that the σ parameter is greater than zero, otherwise calls <a class="link" href="../../error_handling.html#math_toolkit.error_handling.domain_error">domain_error</a>.
        </p>
<pre class="programlisting"><span class="identifier">RealType</span> <span class="identifier">sigma</span><span class="special">()</span><span class="keyword">const</span><span class="special">;</span>
</pre>
<p>
          Returns the <span class="emphasis"><em>sigma</em></span> parameter of this distribution.
        </p>
<h5>
<a name="math_toolkit.dist_ref.dists.rayleigh.h2"></a>
          <span class="phrase"><a name="math_toolkit.dist_ref.dists.rayleigh.non_member_accessors"></a></span><a class="link" href="rayleigh.html#math_toolkit.dist_ref.dists.rayleigh.non_member_accessors">Non-member
          Accessors</a>
        </h5>
<p>
          All the <a class="link" href="../nmp.html" title="Non-Member Properties">usual non-member accessor
          functions</a> that are generic to all distributions are supported:
          <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.cdf">Cumulative Distribution Function</a>,
          <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.pdf">Probability Density Function</a>,
          <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.quantile">Quantile</a>, <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.hazard">Hazard Function</a>, <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.chf">Cumulative Hazard Function</a>,
          <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.mean">mean</a>, <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.median">median</a>,
          <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.mode">mode</a>, <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.variance">variance</a>,
          <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.sd">standard deviation</a>,
          <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.skewness">skewness</a>, <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.kurtosis">kurtosis</a>, <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.kurtosis_excess">kurtosis_excess</a>,
          <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.range">range</a> and <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.support">support</a>.
        </p>
<p>
          The domain of the random variable is [0, max_value].
        </p>
<h5>
<a name="math_toolkit.dist_ref.dists.rayleigh.h3"></a>
          <span class="phrase"><a name="math_toolkit.dist_ref.dists.rayleigh.accuracy"></a></span><a class="link" href="rayleigh.html#math_toolkit.dist_ref.dists.rayleigh.accuracy">Accuracy</a>
        </h5>
<p>
          The Rayleigh distribution is implemented in terms of the standard library
          <code class="computeroutput"><span class="identifier">sqrt</span></code> and <code class="computeroutput"><span class="identifier">exp</span></code> and as such should have very low
          error rates. Some constants such as skewness and kurtosis were calculated
          using NTL RR type with 150-bit accuracy, about 50 decimal digits.
        </p>
<h5>
<a name="math_toolkit.dist_ref.dists.rayleigh.h4"></a>
          <span class="phrase"><a name="math_toolkit.dist_ref.dists.rayleigh.implementation"></a></span><a class="link" href="rayleigh.html#math_toolkit.dist_ref.dists.rayleigh.implementation">Implementation</a>
        </h5>
<p>
          In the following table σ is the sigma parameter of the distribution, <span class="emphasis"><em>x</em></span>
          is the random variate, <span class="emphasis"><em>p</em></span> is the probability and <span class="emphasis"><em>q
          = 1-p</em></span>.
        </p>
<div class="informaltable"><table class="table">
<colgroup>
<col>
<col>
</colgroup>
<thead><tr>
<th>
                  <p>
                    Function
                  </p>
                </th>
<th>
                  <p>
                    Implementation Notes
                  </p>
                </th>
</tr></thead>
<tbody>
<tr>
<td>
                  <p>
                    pdf
                  </p>
                </td>
<td>
                  <p>
                    Using the relation: pdf = x * exp(-x<sup>2</sup>)/2 σ<sup>2</sup>
                  </p>
                </td>
</tr>
<tr>
<td>
                  <p>
                    cdf
                  </p>
                </td>
<td>
                  <p>
                    Using the relation: p = 1 - exp(-x<sup>2</sup>/2) σ<sup>2</sup>= -<a class="link" href="../../powers/expm1.html" title="expm1">expm1</a>(-x<sup>2</sup>/2)
                    σ<sup>2</sup>
                  </p>
                </td>
</tr>
<tr>
<td>
                  <p>
                    cdf complement
                  </p>
                </td>
<td>
                  <p>
                    Using the relation: q = exp(-x<sup>2</sup>/ 2) * σ<sup>2</sup>
                  </p>
                </td>
</tr>
<tr>
<td>
                  <p>
                    quantile
                  </p>
                </td>
<td>
                  <p>
                    Using the relation: x = sqrt(-2 * σ <sup>2</sup>) * log(1 - p)) = sqrt(-2
                    * σ <sup>2</sup>) * <a class="link" href="../../powers/log1p.html" title="log1p">log1p</a>(-p))
                  </p>
                </td>
</tr>
<tr>
<td>
                  <p>
                    quantile from the complement
                  </p>
                </td>
<td>
                  <p>
                    Using the relation: x = sqrt(-2 * σ <sup>2</sup>) * log(q))
                  </p>
                </td>
</tr>
<tr>
<td>
                  <p>
                    mean
                  </p>
                </td>
<td>
                  <p>
                    σ * sqrt(π/2)
                  </p>
                </td>
</tr>
<tr>
<td>
                  <p>
                    variance
                  </p>
                </td>
<td>
                  <p>
                    σ<sup>2</sup> * (4 - π/2)
                  </p>
                </td>
</tr>
<tr>
<td>
                  <p>
                    mode
                  </p>
                </td>
<td>
                  <p>
                    σ
                  </p>
                </td>
</tr>
<tr>
<td>
                  <p>
                    skewness
                  </p>
                </td>
<td>
                  <p>
                    Constant from <a href="http://mathworld.wolfram.com/RayleighDistribution.html" target="_top">Weisstein,
                    Eric W. "Weibull Distribution." From MathWorld--A Wolfram
                    Web Resource.</a>
                  </p>
                </td>
</tr>
<tr>
<td>
                  <p>
                    kurtosis
                  </p>
                </td>
<td>
                  <p>
                    Constant from <a href="http://mathworld.wolfram.com/RayleighDistribution.html" target="_top">Weisstein,
                    Eric W. "Weibull Distribution." From MathWorld--A Wolfram
                    Web Resource.</a>
                  </p>
                </td>
</tr>
<tr>
<td>
                  <p>
                    kurtosis excess
                  </p>
                </td>
<td>
                  <p>
                    Constant from <a href="http://mathworld.wolfram.com/RayleighDistribution.html" target="_top">Weisstein,
                    Eric W. "Weibull Distribution." From MathWorld--A Wolfram
                    Web Resource.</a>
                  </p>
                </td>
</tr>
</tbody>
</table></div>
<h5>
<a name="math_toolkit.dist_ref.dists.rayleigh.h5"></a>
          <span class="phrase"><a name="math_toolkit.dist_ref.dists.rayleigh.references"></a></span><a class="link" href="rayleigh.html#math_toolkit.dist_ref.dists.rayleigh.references">References</a>
        </h5>
<div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; ">
<li class="listitem">
              <a href="http://en.wikipedia.org/wiki/Rayleigh_distribution" target="_top">http://en.wikipedia.org/wiki/Rayleigh_distribution</a>
            </li>
<li class="listitem">
              <a href="http://mathworld.wolfram.com/RayleighDistribution.html" target="_top">Weisstein,
              Eric W. "Rayleigh Distribution." From MathWorld--A Wolfram
              Web Resource.</a>
            </li>
</ul></div>
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